Prologue Some ideas about strange attractors

  • L. Garrido
  • C. Simó
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 179)


For two-dimensional diffeomorphisms or flows reducing essentially to them the evolution of S.A. can be described geometrically using bifurcations, homoclinic and heteroclinic points. However, many questions are left open:
  1. 1)

    Prediction of values of the parameters for which a S.A. appears or is suddenly destroyed.

  2. 2)

    Existence of invariant measures on the S.A. Ergodic or mixing properties of the diffeomorphism restricted to the S.A., with respect to this measure.

  3. 3)

    Examination of the geometry of the S.A. for higher dimensions. Mechanisms producing or destroying S.A. in this case: Study of homo/heteroclinic points of normally hyperbolic invariant or periodic objects.


We strongly recommend to look for the geometric structure in physical or numerical experiments. It seems to us that without this knowledge one cannot get a really deep insight in the problem of S.A.


Periodic Orbit Invariant Measure Chaotic Behavior Invariant Manifold Unstable Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Abraham, R., Shaw, C.:“Dynamics: The Geometry of Behavior”, vol. 1 Aerial Press Inc. 1982 ( following volumes to appear.)Google Scholar
  2. 2.
    Afraîmoviĉ, V.S., Šil'nikov, L.P.: “On some global bifurcations connected with the disappearance of a fixed point of saddle-node type”, Soviet Math. Doklady 15, 1761–1765 (1974).Google Scholar
  3. 3.
    Afraîmoviĉ, V.S., Bykov, V.V., Šil'nikov, L.P.: “On the appearance and structure of the Lorenz-attractor”, Dokl. Akad. Nauk. USSR 234, 336–339 (1977).Google Scholar
  4. 4.
    Aizawa, Y., Shimada, I.: “The Wandering motion on the Lorenz surface”, Prog. Theor. Phys. 57, 2146–2147 (1977)Google Scholar
  5. 5.
    Allgower, E.L., Glashoff, K., Peitgen, H.O. (ed): “Numerical solution of nonlinear equations”, Lect. Notes in Math. 878, Springer (1981).Google Scholar
  6. 6.
    Arneodo, A., Coullet, P., Tresser, C.: “Occurrence of strange attractors in 3-dimensional Volterra equations”, Phys. Lett. 79A, 259–263 (1980).Google Scholar
  7. 7.
    Arneodo, A., Coullet, P., Tresser, C.: “A possible new mechanism for the onset of turbulence”, Phys. Lett. 81A, 197–201 (1981).Google Scholar
  8. 8.
    Arneodo, A., Coullet, P., Tresser, C.: “Possible new strange a.ttractors with spiral structure”, Commun. Math. Phys. 79, 573–579, (1981).Google Scholar
  9. 9.
    Arneodo, A., Coullet, P., Tresser, C.: “Oscillators with Chaotic Behavior: An illustration of a theorem of Šil'nikov”, J. Stat. Phys. 27, 171–182 (1982).Google Scholar
  10. 10.
    Arneodo, A., Coullet, P., Peyraud, J., Tresser, C.: “Strange attractors in Volterra equations for species in competition”, J. Math. Biology 14, 153–157 (1982).Google Scholar
  11. 11.
    Arneodo, A.,Coullet, P., Tresser, C.: “On the relevance of period-doubling cascades at the onset of turbulence”, to appear in Physica D.Google Scholar
  12. 12.
    Arneodo, A.,Coullet, P., Libchaber, A., Maurer, J., D'Humières, D., Tresser, C.: “About the observation of the uncompleted cascade in a Rayleigh-Bénard experiment”, to appear in Physica D.Google Scholar
  13. 13.
    Arnold, V.I.: “Chapitres Supplémentaires de la Théorie des Equations Differentielles Ordinaires”, Editions MIR, Moscow, 1980.Google Scholar
  14. 14.
    Aronson, D.G., Chory, M.A., Hall, G.R., McGehee, R.P.: “Bifurcations from an invariant circle for two parameter families of maps of the Plane: A computer-assisted study”, Commun. in Math. Phys. 83, 303–354 (1982).Google Scholar
  15. 15.
    Atten, P., Lacroix, J.C., Malraison, B.: “Chaotic motion in a Coulomb force driven instability: Large aspect ratio experiments”. Phys. Lett. 77A, 255–258 (1980).Google Scholar
  16. 16.
    Auslander, J., Yorke, J.: “Interval maps, factors of maps and chaos”, Tohoku Math. J. 32, 177–188 (1980).Google Scholar
  17. 17.
    Bernard, P., Ratiu, T. (ed): “Turbulence Seminar”, Lect. Notes in Math. 615, Springer (1977).Google Scholar
  18. 18.
    Bountis, T.C.: “Period doubling and universality in conservative systems”, Physica D (to appear).Google Scholar
  19. 19.
    Bowen, R., Ruelle, D.: “The ergodic theory of axiom A diffeomorphisms”, Inventiones Math. 29, 181–202 (1975).Google Scholar
  20. 20.
    Bowen, R.: “A horseshoe with positive measure”, Inventiones Math. 29, 203–204 (1975).Google Scholar
  21. 21.
    Bowen, R.: “On axiom A diffeomorphism”, A.M.S., C.B.M.S. Regional Conference 35 (1978).Google Scholar
  22. 22.
    Bowen, R.: “Invariant measures for Markov maps of the interval”, Comm. Math. Phys. 69, 1–17 (1979).Google Scholar
  23. 23.
    Baivé, D., Franceschini, V.: “Symmetry breaking on a model of five-mode truncated Navier-Stokes equations”, J. Stat. Phys. 26, 471–484 (1981).Google Scholar
  24. 24.
    Bridges, R., Rowlands, G.: “On the analytic form of some strange attractors”, Physical Lett. 63A, 189–190 (1977).Google Scholar
  25. 25.
    Campanino, M., Epstein, M.: “On the existence of Feigenbaum's fixed point”, Comm. Math. Phys. 79, 261–302 (1981).Google Scholar
  26. 26.
    Chang, Shau-Jin, Wortis, M., Wright, J.A.: “Iterative properties of a one dimensional quartic map: Critical lines and tricritical behavior”, Physical Review 24, 2669–2684 (1981).Google Scholar
  27. 27.
    Chillingworth, D.R.J., Holmes, R.J.: “Dynamical system and the models for reversals of the Earth's magnetic field”, Math. Geology 12, 41–59 (1980).Google Scholar
  28. 28.
    Chirikov, B.V., Izraelev, F.M.: “Some numerical experiments with a nonlinear mapping: Stochastic component”, in “Transformations ponctuelles et leurs applications”, 409–428, Colloques Internationaux du C.N.R.S. 229, and "Degeneration of turbulence in simple systems”, Physica D 2, 30–37 (1981).Google Scholar
  29. 29.
    Clerc, R., Hartmann, Ch.: “Bifurcation mechanism of a second order recurrence leading to the appearance of a particular strange attractor”, preprint, Univ. of Toulouse.Google Scholar
  30. 30.
    Collet, P., Eckmann, J.P.: “On the abundance of aperiodic behavior for maps of the interval”, Comm. Math. Phys. 73, 115–160 (1980).Google Scholar
  31. 31.
    Collet, P., Eckmann, J.P., Lanford, O.: “Universal properties of maps of an interval”, Comm. Math. Phys. 76, 211–254 (1980).Google Scholar
  32. 32.
    Collet, P., Eckmann, J.P.: “Iterated maps on the interval as dynamical systems”, Birkhäuser, Boston, 1980.Google Scholar
  33. 33.
    Collet, P., Eckmann, J.P., Koch, H.: “Period doubling bifurcations for families of maps on Rn”, J. Stat. Phys. 25 (1981).Google Scholar
  34. 34.
    Collet, P., Crutchfield, J.P., Eckmann, J.P.: “Computing the topological entropy of maps”, to appear in Physica D.Google Scholar
  35. 35.
    Cook, A., Roberts, D.: “The Rikitake two-disc dynamo system”, Proc. Cambridge Philos. Soc. 68, 547–569 (1970).Google Scholar
  36. 36.
    Coullet, P., Tresser, C.: “Critical transition to “Stochasticity” for some dynamical systems”, J. de Physique Lett. 41, L 255 (1980).Google Scholar
  37. 37.
    Crutchfield, J.P.: “Prediction and stability in classical mechanics, University of Santa Cruz Thesis, 1979.Google Scholar
  38. 38.
    Crutchfield, J.P., Farmer, J.D., Packard, N.H., Shaw, R.S., Jones, G., Donnelly, R.: “Power Spectral Analysis of a Dynamical System”, Phys. Lett. 76A, 1 (1980).Google Scholar
  39. 39.
    Crutchfield, J.P., Huberman, B.A.: “Fluctuations and the onset of chaos”, Phys. Lett. 77A, 407 (1980).Google Scholar
  40. 40.
    Crutchfield, J.P., Farmer, J.D., Huberman, B.A.: “Fluctuations and simple chaotic dynamics”, to appear in Physics Reports.Google Scholar
  41. 41.
    Crutchfield, J.P., Nauenberg, M., Rudnick, J.: “Scaling for external noise at the onset of chaos”, Phys. Rev. Lett. 46, 933 (1981).Google Scholar
  42. 42.
    Crutchfield, J.P., Packard, N.H.: “Symbolic dynamics of one-dimensional maps: entropies, finite precision and noise”, to appear in Intl. J. Theor. Phys.Google Scholar
  43. 43.
    Curry, J.: “A generalized Lorenz system”, Comm. Math. Phys. 60, 193–204 (1978).Google Scholar
  44. 44.
    Curry, J.: “On the Hénon transformation”, Comm. Math. Phys. 68, 129–140 (1979).Google Scholar
  45. 45.
    Curry, J.: “On computing the, entropy of the Hénon attractor”, J. Stat. Phys. 26, 683–695 (1981).Google Scholar
  46. 46.
    Daido, H.: “Analytical conditions for the appearance of homoclinic and heteroclinic points of a 2-dimensional mapping”, Prog. Theor. Phys. 63, 1190–1201 (1980).Google Scholar
  47. 47.
    Daido, N.: “Universal relation of a band-splitting sequence to a preceding period doubling one”, Phys. Lett. 86A, 259–262 (1981).Google Scholar
  48. 48.
    Dell'Antonio, G., Doplicher, S., Jona-Lasinio, G. (ed.): “Mathematical problems in theoretical physics”, Lect. Notes in Physics 80, Springer (1978).Google Scholar
  49. 49.
    Derrida, B., Pomeau, Y.: “Feigenbaum's ratios of 2-dimensional area preserving maps”, Phys. Lett. 80A, 217–219 (1980).Google Scholar
  50. 50.
    Donnelly, R.J., Park, K., Shaw, R.S., Walden, R.W.: “Early nonperiodic transition in Couette flow”, Phys. Rev. Lett. 44, 987 (1980).Google Scholar
  51. 51.
    Douady, A., Oesterlé, J.: “Dimension de Hausdorff des attracteurs”, C.R. Acad. Sci. Paris, 290, 1135–1138 (1980).Google Scholar
  52. 52.
    Easton, R.: “A topological conjugacy invariant involving homoclinic points for diffeomorphisms of two-manifolds”, preprint, Univ. of Boulder, Colorado.Google Scholar
  53. 53.
    Eckmann, J.P.: “Roads to turbulence in dissipative dynamical systems”, Rev. Mod. Phys. 53, 643–654 (1981).CrossRefGoogle Scholar
  54. 54.
    Eckmann, J.P.: “Renormalization group analysis of some highly bifurcated families”, preprint, Univ. of Genève.Google Scholar
  55. 55.
    Eckmann, J.P.: “Intermittency in the presence of noise”, J. Phys. A 14, 3153–3168 (1981).Google Scholar
  56. 56.
    Eckmann, J.P.: “A note on the power spectrum of the iterates of Feigenbaum's function”, Commun. Math. Phys. 81, 261–265 (1981).Google Scholar
  57. 57.
    Farmer, J.D.: “Spectral broadening of period-doubling bifurcation sequences”, Phys. Rev. Lett. 47, 179 (1981).Google Scholar
  58. 58.
    Farmer, J.D.: “Order within chaos”, Univ. of Santa Cruz, Ph.D. Thesis (1981).Google Scholar
  59. 59.
    Farmer, J.D.: “Chaotic attractors of an infinite-dimensional dynamical system”, Physica D 4, 366–393 (1982).Google Scholar
  60. 60.
    Farmer, J.D. “Information dimension and the probabilistic structure of chaos”, to appear in Z. Naturforschung.Google Scholar
  61. 61.
    Farmer, J.D., Ott, E., Yorke, J.A.: “The dimension of chaotic attractors”, to appear in Physica D.Google Scholar
  62. 62.
    Farrell, F.T., Jones, L.E.: “New attractors in hyperbolic dynamics, J. Diff. Geometry 15, 107–133 (1980).Google Scholar
  63. 63.
    Fatou, P.: “Sur les equations fonctionelles”, Bull. Soc. Math. de France, 47, 161–270 (1919), 48, 33–95 & 208–314 (1920).Google Scholar
  64. 64.
    Feigenbaum, M.: “Quantitative universality for a class of nonlinear transformations”, J. Stat. Phys. 19, 25–52 (1978).CrossRefGoogle Scholar
  65. 65.
    Feigenbaum, M.: “The onset spectrum of turbulence”, Phys. Lett. 74A, 375 (1979)Google Scholar
  66. 66.
    Feigenbaum, M.: “The transition to aperiodic behavior in turbulent systems”, Commun. Math. Phys. 77, 65–86 (1980).Google Scholar
  67. 67.
    Feit, S.: “Characteristic exponents and strange attractors”, Commun. Math. Phys. 61, 249–260 (1978).Google Scholar
  68. 68.
    Franceschini, V.: “Feigenbaum sequence of bifurcations in the Lorenz model”, J. Stat. Phys. 22, 397–407 (1980).Google Scholar
  69. 69.
    Franceschini, V., Russo, L.: “Stable and unstable manifolds of the Hénon mapping”, preprint, Univ. of Modena.Google Scholar
  70. 70.
    Franceschini, V.: “Two models of truncated Navier-Stokes equations on a two-dimensional torus”, Phys. of Fluids (to appear).Google Scholar
  71. 71.
    Franceschini, V.: “Truncated Navier-Stokes equations on a two-dimensional torus”, preprint, Los Alamos.Google Scholar
  72. 72.
    Franceschini, V.: “Bifurcation of tori and phase-locking in a dissipative system of differential equations”, preprint, Los Alamos.Google Scholar
  73. 73.
    Frederickson, P., Kaplan, J.L., Yorke, E.D., Yorke, J.: “The Lyapunov dimension of strange attractors”, J. Diff. Equations (to appear).Google Scholar
  74. 74.
    Fujisaka, H., Yamada, T.: “Limit cycles and chaos in realistic models of the Belousov-Zhabotinskii reaction system, Z. Physik B, 37, 265–275 (1980).Google Scholar
  75. 75.
    Garrido, L. (ed.): “Systems far from equilibrium”, Lect. Notes in Physics 132 (1980).Google Scholar
  76. 76.
    Gollub, J.P. Swinney, H.L.: “Onset of turbulence in a rotating fluid”, Phys. Rev. Lett. 35, 921 (1975).Google Scholar
  77. 77.
    Grebogi, C., Ott, E., Yorke, J.A.: “Chaotic attractors in crisis”, Phys. Rev. Lett. 48, 1507–1510 (1982).Google Scholar
  78. 78.
    Grebogi, C., Ott, E., Yorke, J.A.: “Crisis, sudden changes in chaotic attractors and transient chaos”, preprint, Univ. of Maryland.Google Scholar
  79. 79.
    Grmela, M., Marsden, J.E. (ed.): “Global analysis”, Lect. Notes in Math. 755, Springer (1979).Google Scholar
  80. 80.
    Grossmann, S., Thomae, S.: “Invariant distributions and stationary correlation functions”, Z. Naturforschung 32A, 1353–1365 (1977).Google Scholar
  81. 81.
    Guckenheimer, J.: “Bifurcations of maps of the interval”, Inventiones math. 39, 165–178 (1977).Google Scholar
  82. 82.
    Guckenheimer, J.: “Sensitive dependence on initial conditions for one-dimensional maps”, Commun. Math. Phys. 70, 133–160 (1979).Google Scholar
  83. 83.
    Guckenheimer, J., Moser, J., Newhouse, S.: “Dynamical Systems”, Birkhäuser, Boston, 1980.Google Scholar
  84. 84.
    Guckenheimer, J., Williams, R.: “Structural stability of Lorenz attractors”, Pub. Math. I.H.E.S. 50, 60–72 (1980).Google Scholar
  85. 85.
    Gumowski, I., Mira, C.: “Dynamique Chaotique”, Editions Cepadues, Toulouse, 1980.Google Scholar
  86. 86.
    Gumowski, I., Mira, C.: “Recurrence and discrete dynamical systems”, Lect. Notes in Math. 809, Springer, 1980.Google Scholar
  87. 87.
    Gurel, O., Rössler, O.E. (ed.): “Bifurcation theory and applications in scientific disciplines”, Annals New York Acad. Sci. 316, (1979).Google Scholar
  88. 88.
    Haken, H.: “Synergetics, An Introduction”, 2nd ed. Springer series in Synergetics, 1, (1978).Google Scholar
  89. 89.
    Haken, H. (ed.): “Synergetics, A Workshop”, Springer series in Synergetics 2, (1977).Google Scholar
  90. 90.
    Haken, H., Wunderlin, A.: “New interpretation and size of strange attractors of the Lorenz model of turbulence”, Phys. Lett. 62A, 133–134 (1977).Google Scholar
  91. 91.
    Haken, H. (ed.): “Chaos and order in Nature”, Springer (1981).Google Scholar
  92. 92.
    Haken, H. (ed.): “Evolution of ordered and chaotic patterns in systems treated by the natural sciences and mathematics”, Springer (1982) (to appear).Google Scholar
  93. 93.
    Helleman, R.H.G.: “Self-generated chaotic behavior in nonlinear mechanics”, in “Fundamental Problems in Statistical Mechanics V”, Ed.: E.G.D. Cohen, 165–233, North Holland (1980).Google Scholar
  94. 94.
    Helleman, R.H.G. (ed.): “Nonlinear dynamics”, Annals New York Acad. Sci. 357 (1980).Google Scholar
  95. 95.
    Helleman, R.H.G., Iooss, G. (ed.): “Chaotic behavior in deterministic systems”, North-Holland, (1982) (to appear).Google Scholar
  96. 96.
    Hénon, M.: “A two-dimensional mapping with a strange attractor”, Commun. Math. Phys. 50, 69–77 (1976).Google Scholar
  97. 97.
    Herring, C., Huberman, B.A.: “Dislocation motion and solid-state turbulence”, Appl. Phys. Lett. 36, 975–977 (1980).Google Scholar
  98. 98.
    Hirsch, M.W., Pugh, C.C., Shub, M.: “Invariant manifolds”, Lect. Notes in Math. 583, Springer (1977).Google Scholar
  99. 99.
    Hitzl, D.L.: “Numerical determination of the capture/escape boundary for the Hénon attractor”, preprint Lockheed, Palo Alto (1981).Google Scholar
  100. 100.
    Holmes, P.J.: “Strange phenomena in dynamical systems and their physical implications”, Appl. Math. Modelling, 1, 362–366 (1977).Google Scholar
  101. 101.
    Holmes, P.J.: “A nonlinear oscillator with a stran e attractor”, Phil. Trans. Roy. Soc. London, Ser. A 292, 419–448 (1979).Google Scholar
  102. 102.
    Holmes, P.J.: “Averaging and chaotic motions in forced oscillations”, SIAM J. Appl. Math. 38, 65–80 (1980).Google Scholar
  103. 103.
    Holmes, P.J., Moon, F.C.: “A magneto-elastic strange attractor”, J. Sound vibration 65, 275–296 (1979).Google Scholar
  104. 104.
    Holmes, P.J. (ed.): “New approaches to nonlinear problems in dynamics”, SIAM, Philadelphia (1980).Google Scholar
  105. 105.
    Hoppenstedt, F.C. (ed.): “Nonlinear oscillations in Biology”, A.M.S. Lectures in Appl. Math. 17, Providence (R.I.) (1979).Google Scholar
  106. 106.
    Huberman, B.A., Crutchfield, J.P.: “Chaotic states of anharmonic systems in periodic fields”, Phys. Rev. Lett. 43, 1743–1747 (1979).Google Scholar
  107. 107.
    Huberman, B.A., Crutchfield, J.P., Packard, N.H.: “Noise phenomena in Josephson junctions”, Appl. Phys. Lett. 37, 750–752 (1980).Google Scholar
  108. 108.
    Huberman, B.A., Rudnick, J.: “Scaling behavior of chaotic flows” Phys. Rev. Lett. 45, 154–156 (1980).Google Scholar
  109. 109.
    Hurley, M.: “Attractors: Persistence and density of their basins”, Trans. Amer. Math. Soc. 269, 247–271 (1982).Google Scholar
  110. 110.
    Izraelev, F.M., Rabinovich, M.I., Ugodnikov, A.D.: “Approximate description of three dimensional dissipative systems with stochastic behavior”, Phys. Lett. 86A, 321–325 (1981).Google Scholar
  111. 111.
    Jakobson, M.V.: “Absolutely continuous invariant measures for one-parameter families of one-dimensional maps”. Commun. Math. Phys. 81, 39–88 (1981).Google Scholar
  112. 112.
    Jeffreis, C., Péerez, J.: “Direct observation of crisis of the chaotic attractor in a nonlinear oscillation”, preprint, Univ. of Berkeley (1982).Google Scholar
  113. 113.
    Jonker, L., Rand, D.: “Bifurcations in one-dimension, I, II,” Inventiones math. 62, 347–365 and 63, 1–15 (1981).Google Scholar
  114. 114.
    Jorna, S.(ed.): “Topics in Nonlinear Dynamics”, Amer. Inst. Phys. Conf. Proc. 46, (1978).Google Scholar
  115. 115.
    Julia, G.: “Mémoire sur l'itération des fonctions rationelles”, J. de Math. sér. 7, 4, 47–245 (1918).Google Scholar
  116. 116.
    Kai, T., Tomita, K.: “Stroboscopic phase-portrait of a forced nonlinear oscillator”, Prog. Theor. Phys. (to appear).Google Scholar
  117. 117.
    Kaplan, J.L., Yorke, J.A.: “Preturbulence: A regime observed in a fluid flow model of Lorenz”, Commun. Math. Phys. 67, 93–108, (1979).Google Scholar
  118. 118.
    Keener, J.P.: “Chaotic cardiac dynamics”, in Lect. in Appl. Math. 19, 299–325, “Mathematical Aspects of Physiology”, ed. F.C. Hoppensteadt, AMS (1981).Google Scholar
  119. 119.
    Kidachi, H.: “On a chaos as a mode interaction phase”, Prog. Theor. Phys. 65, 1584–1594 (1981).Google Scholar
  120. 120.
    Knobloch, E.: “Chaos in the segmented disc dynamo”, Phys. Lett. 82A, 439–440 (1981).Google Scholar
  121. 121.
    Lasota, A., Yorke, J.: “On the existence of invariant measures for piecewise monotonic transformations”, Trans. Amer. Math. Soc. 184, 481–488 (1973).Google Scholar
  122. 122.
    Laval, G., Gresillon, D. (ed.): “Intrinsic stochasticity in plasmas”, International Workshop, Cargèse, Les Editions de Physique, Orsay (1979).Google Scholar
  123. 123.
    Leipnick, R.B., Newton, T.A.: “Double strange attractors in rigid body motion with linear feedback control”, Phys. Lett. 86A, 63–67 (1981).Google Scholar
  124. 124.
    Leven, R.W., Koch, B.D.: “Chaotic behavior of a parametrically excited damped pendulum”, Phys. Lett. 86A, 71–74 (1981).Google Scholar
  125. 125.
    Levi, M.: “Qualitative analysis of the periodically forced relaxation oscillators”, Mem. A.M.S. 244, (1981).Google Scholar
  126. 126.
    Li, T.Y., Yorke, J.A.: “Period three implies chaos”, Americ. Math. Monthly, 82, 985–992 (1975).Google Scholar
  127. 127.
    Li, T.Y., Misiurewicz, M., Pianigiani, G., Yorke, J.: “Odd chaos”, Phys. Lett. 87A, 271–273 (1982).Google Scholar
  128. 128.
    Libchaber, A., Maurer, J.: “Une expérience de Bénard-Rayleigh de géometrie réduite; multiplication, accrochage et démultiplication de fréquences”, J. de Physique 41, (Coll. C3), 51–56 (1980).Google Scholar
  129. 129.
    Lopes, A.D.: “An example of interpolation of an attractor”, preprint, Inst. Mat. Porto Alegre.Google Scholar
  130. 130.
    Lorenz, E.N.: “Deterministic nonperiodic flow”, J. Atmosph. Sci. 20, 130–141 (1963).Google Scholar
  131. 131.
    Lozi, R.: “Un attracteur étrange (?) du type attracteur de Hénon”, J. de Physique 39 (Coll. C5), 9–10 (1978).Google Scholar
  132. 132.
    Lozi, R.: “Sur un modéle mathématique de suite de bifurcations de motifs dans la réaction de Belousov-Zhobotinsky”, C.R. Acad. Sci. Paris 294, 21–26 (1982).Google Scholar
  133. 133.
    L'vov, V.S., Predtechansky, A.A.: “On Landau and stochastic pictures in the problem of transition to turbulence, Physica D, 2 38–51 (1981).Google Scholar
  134. 134.
    Mackey, M.C., Glass, L.: “Oscillators and chaos in physiological control systems”, Science 197, 287–289 (1977).PubMedGoogle Scholar
  135. 135.
    Manneville, P.: “Intermittency in dissipative dynamical systems”, Phys. Lett. 79A, 33–35 (1980).Google Scholar
  136. 136.
    Manneville, P., Pomeau, Y.: “Different ways to turbulence in dissipative dynamical systems”, Physica D, 1, 219–226 (1980).Google Scholar
  137. 137.
    Manneville, P.: “Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems”, J. Physique 41, 1235–1243 (1980).Google Scholar
  138. 138.
    Markley, N.G., Martin, J.C., Perrizo, W. (ed.): “The structure of attractors in dynamical systems”, Lect. Notes in Math. 668, Springer, (1978).Google Scholar
  139. 139.
    Marotto, F.R.: “Chaotic behavior in the Hénon mapping”, Commun. Math. Phys. 68, 187–194 (1979).Google Scholar
  140. 140.
    Marsden, J.E., McCracken, M.: “The Hopf bifurcation and its applications”, Appl. Math. Sci. 19, Springer (1976).Google Scholar
  141. 141.
    Marzec, C.J., Spiegel, E.A.: “Ordinary differential equations with strange attractors”, SIAM J. Appl. Math. 38, 387–421 (1980).Google Scholar
  142. 142.
    May, R., Oster, G.: “Bifurcations and dynamic complexity in simple ecological models”, The Amer. Natur. 110, 573–599 (1976).Google Scholar
  143. 143.
    Mayer-Kress, G., Haken, H.: “Intermittent behavior of the logistic system”, Phys. Lett. 82A, 151–155 (1981).Google Scholar
  144. 144.
    McLaughlin, J., Martin, P.: “Transition to turbulence in a statically stressed fluid system”, Phys. Rev. A 12, 186–203 (1975).Google Scholar
  145. 145.
    Metropolis, M., Stein, M.L., Stein, P.R.: “On finite limit sets for transformations of the unit interval”, J. Combinatorial Theory 15, 25–44 (1973).Google Scholar
  146. 146.
    Milnor, J., Thurston, W.: “On iterated maps of the interval”, preprint, Univ. of Princeton.Google Scholar
  147. 147.
    Mira, C.: “Accumulation de bifurcations et “structures boîtes emboîtées dans les recurrences et transformations ponctuelles”, VII ICNO, Berlin (1975).Google Scholar
  148. 148.
    Misiurewicz, M., Swecz, B.: “Existence of a homoclinic point for the Hénon map”, Commun. Math. Phys. 75, 285–291 (1980).Google Scholar
  149. 149.
    Myrberg, P.J.: “Iteration der reellen Polynome zweiten Grades, I, II, II”, Ann. Acad. Sci. Fenn. 256A, 1–10 (1958); 268A, 1–10 (1959), 336A, 1–10 (1963).Google Scholar
  150. 150.
    Myrberg, P.J.: “Iteration der Polynome mit reellen Koeffizienten” Ann. Acad. Sci. Fenn. 374A, 1–18 (1965).Google Scholar
  151. 151.
    Newhouse, S.: “Diffeomorphisms with infinitely many sinks”, Topology 12, 9–18 (1974).Google Scholar
  152. 152.
    Nitecki, Z., Robinson, C. (ed.): “Global theory of dynamical systems”, Lect. Notes in Math. 819, Springer (1980).Google Scholar
  153. 153.
    Ogura, H., Ueda, Y., Yoshida, Y.: “Periodic stationarity of a chaotic motion in the system governed by Duffing's equation”, Prog. Theor. Phys. 66, 2280–2283 (1981).Google Scholar
  154. 154.
    Oono, Y., Osikawa, M.: “Chaos in nonlinear differential equations, I”, Prog. Theor. Phys. 64, 54–67 (1980).Google Scholar
  155. 155.
    Oseledec, V.I.: “A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems”, Trans. Moscow Math. Soc. 19, 197–231 (1968).Google Scholar
  156. 156.
    Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S.: “Geometry from a time series”, Phys. Rev. Lett. 45, 712 (1980).Google Scholar
  157. 157.
    Peitgen, H.O., Walther, H.O. (ed.): “Functional differential equations and approximation of fixed points”, Lect. Notes in Math. 730, Springer (1979).Google Scholar
  158. 158.
    Pesin, Ya. B.: “Characteristic Lyapunov exponents and smooth ergodic theory”, Russ. Math. Surveys, 32, 55–115 (1977).Google Scholar
  159. 159.
    Peters, H.: “Chaotic behavior of nonlinear differential-delay equations”, preprint, Univ. of Bremen.Google Scholar
  160. 160.
    Pikowsky, A.S., Rabinovich, M.I.: “Stochastic oscillations in dissipative systems”. Physica D 2, 8–24 (1981).Google Scholar
  161. 161.
    Pixton, D.: “Planar homoclinic points”, J. of Diff. Eq. 44, 365–382 (1982).Google Scholar
  162. 162.
    Plykin, R.: “Sources and sinks for A-diffeomorphisms”, Math. Sb. 23, 233–253 (1974).Google Scholar
  163. 163.
    Pomeau, Y., Manneville, P.: “Intermittent transition to turbulence in dissipative dynamical systems”, Commun. Math. Phys. 74, 189–197 (1980).CrossRefGoogle Scholar
  164. 164.
    Pounder, J.R., Rogers, T.D.: “The geometry of chaos:dynamics of a nonlinear second order difference equations”, Bull. Math. Biol. 42, 551–597 (1980).Google Scholar
  165. 165.
    Rand, D.A., Young, L.S. (ed.): “Dynamical systems and turbulence, Warwick 1980”, Lect. Notes in Math. 898, Springer (1981).Google Scholar
  166. 166.
    Roux, J.C., Rossi, A., Bachelart, S., Vidal, C.: “Representation of a strange attractor from an experimental study of chemical turbulence”, Phys. Lett. 77A, 391–393 (1980).Google Scholar
  167. 167.
    Ruelle, D., Takens, F.: “On the nature of turbulence”, Commun. Math. Phys. 20, 167–192 (1971).CrossRefGoogle Scholar
  168. 168.
    Ruelle, D.: “A measure associated to axiom A attractors”, Amer. J. of Math. 98, 619–654 (1976).Google Scholar
  169. 169.
    Ruelle, D.: “Applications conservant une mésure absolument continue par rapport â dx sur [o,1]”, Comm. Math. Phys. 55, 47–51 (1977).Google Scholar
  170. 170.
    Ruelle, D.: “Strange attractors”, The Mathematical Intelligenter 2, 126–137 (1980).Google Scholar
  171. 171.
    Ruelle, D.: “Small random perturbations of dynamical systems and the definition of attractors”, Commun. Math. Phys. 82, 137–151 (1981).Google Scholar
  172. 172.
    Ruelle, D.: “Do there exist turbulent crystals?”, preprint, I.H.E.S.Google Scholar
  173. 173.
    Šarkovskii, A.N.: “Coexistence of cycles of a continuous map of a line into itself”, Ukr. Mat. Z. 16, 61–71 (1964).Google Scholar
  174. 174.
    Scholz, H.J., Yamada, T., Brand, H., Graham, R.: “Intermittency and chaos in a laser system with modulated inversion”, Phys. Lett. 82A, 321–323 (1981).Google Scholar
  175. 175.
    Shaw, R.S.: “Strange attractors, chaotic behavior, and information flow”, Z. Naturforschung 36A, 80 (1981).Google Scholar
  176. 176.
    Shaw, R.S.: “On the predictabilit of mechanical systems”, Univ. of Santa Cruz, Ph.D. Thesis (1980).Google Scholar
  177. 177.
    Shaw, R.S., Anderek, C.D., Reith, L.A., Swinney, M.L.: “Nonlinear superposition of traveling waves in circular Couette flow”, Phys. Rev. Lett. (to appear).Google Scholar
  178. 178.
    Shimada, I., Nagashima, T.: “The iterative transition phenomenon between periodic and turbulent states in a dissipative dynamical system”, Prog. Theor. Phys. 59, 1033–1035 (1978).Google Scholar
  179. 179.
    Shimada, I.: “Gibbsian distribution on the Lorenz attractor”, Prog. Theor. Phys. 62, 61–69 (1979).Google Scholar
  180. 180.
    Shimizu, T. ”Asymptotic form of a strange attractor”, Phys. Lett. 84A, 85 (1981).Google Scholar
  181. 181.
    Simó, C.: “On the Hénon-Pomeau attractor”, J. Stat. Phys. 21, 465–493 21 (1979)Google Scholar
  182. 182.
    Simonov, A.A.: “An investigation of bifurcations in some dynamical systems by the methods of symbolic dynamics”, Soviet Math. Dok. 19, 759–763 (1978).Google Scholar
  183. 183.
    Sinai, J.G. Vul, E.B.: “H perbolicity conditions for the Lorenz model”, Physica D, 2, 3–7 (1981).Google Scholar
  184. 184.
    Smale, S.: “Diffeomorphisms with many periodical points”, in Differential and Combinatorial Topology, 63–80, Princeton Univ. Press (1965).Google Scholar
  185. 185.
    Smale, S.: “Differentiable dynamical systems”, Bull. Amer. Math. Soc. 73, 747–817 (1967).Google Scholar
  186. 186.
    Smale, S. (ed.): “Global analysis”, Proc. Sympos. Pure Math. 14, Amer. Math. Soc. (1970).Google Scholar
  187. 187.
    Steeb, W.H., Kunick, A.: “Lagrange functions of a class of dynamical systems with limit cycle and chaotic behavior”, Phys. Rev. A 25, 2889–2892 (1982).Google Scholar
  188. 188.
    Steeb, W.H.: “Continuous symmetries of the Lorenz and the Rikitake two-disc dynamo system”, J. Phys. A 15, L 389–390 (1982).Google Scholar
  189. 189.
    Steeb, W.H., Erig, W., Kunick, A.: “Chaotic behavior and limit cycle behavior of anharmonic systems with periodic external perturbations”, preprint, Univ. of Paderborn.Google Scholar
  190. 190.
    Steeb, W.H., Kunick, A.: “On the Painlevé property of anharmonic systems with an external period field”, preprint, Univ. of Paderborn.Google Scholar
  191. 191.
    Stefan, P.: “A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line”, Commun. Math. Phys. 54, 237–248 (1977).Google Scholar
  192. 192.
    Swinney, H.L., Gollub, J.P.: “The transition to turbulence”, Physics Today 31, 41–49 (1978).Google Scholar
  193. 193.
    Takeyama, K.: “Dynamics of the Lorenz model of convective instabilities II”, Prog. Theor. Phys. 63, 91–105 (1980).Google Scholar
  194. 194.
    Tél, T.: “On the construction of stable and unstable manifolds of two-dimensional invertible maps”, preprint, Eötvös Univ.Google Scholar
  195. 195.
    Temam, R. (ed.): “Turbulence and Navier-Stokes Equation”, Lect. Notes in Math. 565, Springer (1976).Google Scholar
  196. 196.
    Testa, J., Held, G.A.: “Period doubling, bifurcations, chaos and periodic windows of the cubic map”, preprint, Univ. of Berkeley.Google Scholar
  197. 197.
    Thomae, S., Grossmann, S.: “Correlations and spectra of periodic chaos generated by the logistic parabola”, J. Stat. Phys. 26, 485–504 (1981).Google Scholar
  198. 198.
    Thurston, W.: “On the geometry and dynamics of diffeomorphisms of surfaces”, preprint, Univ. of Princeton.Google Scholar
  199. 199.
    Tomita, K., Kai, T.: “Stroboscopic phase-portrait and strange attractors”, Phys. Lett. 66A, 91–93 (1978).Google Scholar
  200. 200.
    Tomita, K., Daido, H.: “Possibility of chaotic behavior and multi-basins in forced glycolytic oscillations”, Phys. Lett. 79A, 133–137 (1980).Google Scholar
  201. 201.
    Tomita, K., Kai, T.: “Chaotic response of a nonlinear oscillator” J. Stat. Phys. (to appear).Google Scholar
  202. 202.
    Tomita, K., Tsuda, I.: “Towards an interpretation of Hudson's experiment on the Belousov-Zhabotinskii Reaction: chaos due to delocalization”, Prog. Theor. Phys. 64, 1138–1160 (1980).Google Scholar
  203. 203.
    Tomita, K., Tsuda, I.: “Chaos in the Belousov-Zhabotinskii reaction in a flow system”, Phys. Lett. 71A, 489–492 (1979).Google Scholar
  204. 204.
    Tomita, K.: “Chaotic behavior of deterministic orbits. The problem of turbulent phase”, preprint, Univ. of Kyoto.Google Scholar
  205. 205.
    Tresser, C., Coullet, P., Arneodo, A.: “On the existence of hysteresis in a transition to chaos after a single bifurcation”, J. de Physique Lettres 41, L 243–246 (1980).Google Scholar
  206. 206.
    Tresser, C., Coullet, P., Arneodo, A.: “Topological horseshoe and numerically observed chaotic behavior in the Hénon mapping”, Lett. to the Editor, J. Phys. A13, L 123–127 (1980).Google Scholar
  207. 207.
    Tsuda, I. “Self-similarity in the Belusov-Zhabotinsky reaction” Phys. Lett. 85A, 4 (1981).Google Scholar
  208. 208.
    Ueda, Y., Hayashi, C., Akamatsu, N.: “Computer simulation of nonlinear ordinary differential equations and nonperiodic oscillations”, Electronics and Communications in Japan 56A, 27–34 (1973).Google Scholar
  209. 209.
    Ueda, Y.: “Randomly transitional phenomena in the system governed by Duffing]"s equation”, J. Stat. Phys. 20, 181–196 (1979).Google Scholar
  210. 210.
    Ueda, Y., Akamatsu, N.: “Chaoticaly transitional phenomena in the forced negative-resistance oscillator”, IEEE, Trans. on Circuits and Systems 28, 217–224 (1981).Google Scholar
  211. 211.
    Ulam, S., Von Neumann, J.: “On combinations of stochastic and deterministic processes”, Bull. Amer. Math. Soc. 53, 1120 (1947).Google Scholar
  212. 212.
    Velsen, R.V., Oberman, C.R.: “Statistical properties of chaotic dynamical systems which exhibit strange attractors”, Physica D, 4, 183–196 (1982).Google Scholar
  213. 213.
    Vidal, Ch. et al.: “Étude de la transition vers la turbulence chimique dans la réaction de Belousov-Zhabotinskii”, C.R. Acad. Sci. Paris 289 C, 73–77 (1979).Google Scholar
  214. 214.
    Walther, H.O.: “Homoclinic solutions and chaos in x(t)=f(x(t-1))”, Nonlinear Analysis: Theory, Methods and Appl. 5, 775–788 (1981).Google Scholar
  215. 215.
    Wegmann, K., Rössler, D.: “Different kinds of chaotic oscillations in the Belousov-Zhabotinskii reaction”, Z. Naturforschung 33A, 1179–1183 (1978).Google Scholar
  216. 216.
    Wersinger, J.M., Finn, J.H. Ott, E.: “Bifurcation and “strange” behavior in instability saturation by nonlinear three-wave mode coupling”, Phys. Fluids 23, 1142–1154 (1980).CrossRefGoogle Scholar
  217. 217.
    Willamowski, K.D., Rössler, O.E.: “Irregular oscillations in a realistic abstract quadratic mass action system”, Z. Naturforschung 35A, 317–318 (1980).Google Scholar
  218. 218.
    Williams, R.: “One dimensional nonwandering sets”, Topology 6, 473–487 (1967).CrossRefGoogle Scholar
  219. 219.
    Williams, R.: “The structure of Lorenz attractors”, Pub. Math. I.H.E.S. 50, 73–99 (1980).Google Scholar
  220. 220.
    Young, L.S.: “Capacity of attractors”,Ergod. Th. and Dynam. Syst. 1, 381–388 (1981).Google Scholar
  221. 221.
    Zaslavsky, G.M.: “The simplest case of a strange attractor”, Phys. Letters 69A, 145–147 (1978).Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • L. Garrido
    • 1
  • C. Simó
    • 2
  1. 1.Facultad de FísicaUniversidad de BarcelonaBarcelona-28Spain
  2. 2.Facultat de MatemàtiquesUniversitat de BarcelonaBarcelona-7Spain

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